# Lenz's Law

Lenz's law describes the direction of the induced EMF. The law states that the direction of the induced EMF is such that it opposes the change in the magnetic field that produced it. In other words, the direction of induced EMF will be such that it creates a magnetic field that opposes the original change in the magnetic field.

Induced current create a magnetic field that opposes the change in magnetic flux.

Induced currents are setup inside a metallic object when magnetic flux through it varies. These induced currents are called eddy currents. Eddy currents are an example of Lenz's law.

## Problems from IIT JEE

Problem (IIT JEE 1997): An infinitesimally small bar magnet of dipole moment $\vec{M}$ is pointing and moving with the speed $v$ in the positive $x$ direction. A small closed circular conducting loop of radius $a$ and negligible self-inductance lies in the $y\text{-}z$ plane with its centre at $x=0$, and its axis coinciding with the $x$-axis. Find the force opposing the motion of the magnet, if the resistance of the loop is $R$. Assume that the distance $x$ of the magnet from the centre of the loop is much greater than $a$.

Solution: The magnetic field due to an infinitesimal bar magnet for end-on position is given by, \begin{alignat}{2} & \vec{B}=\frac{\mu_0}{4\pi}\frac{2M}{x^3}\,\hat\imath, \nonumber \end{alignat} where $M$ is magnetic moment and $x$ is the distance from the magnet. Since distance $x$ is much greater than radius~$a$ of the loop, magnetic field can be taken as uniform throughout the loop. For the loop, flux $\phi$, induced emf $e$, induced current $i$ and the magnetic moment $M^\prime$ are given by, \begin{align} \phi&=\vec{B}\cdot\vec{S}\nonumber\\ &=\frac{\mu_0}{4\pi}\frac{2M}{x^3}(\pi a^2)\nonumber\\ &=\frac{\mu_0a^2M}{2x^3} \end{align} \begin{align} e&=-\frac{\mathrm{d}\phi}{\mathrm{d}t}\nonumber\\ &=\frac{3\mu_0 a^2M}{2x^4}\frac{\mathrm{d}x}{\mathrm{d}t}\nonumber\\ &=\frac{3\mu_0 a^2Mv}{2x^4}\nonumber \end{align} \begin{align} i&=\frac{e}{R}=\frac{3\mu_0 a^2Mv}{2Rx^4}, \nonumber\\ M^\prime&=iS=\frac{3\pi\mu_0 a^4Mv}{2Rx^4}. \nonumber \end{align} According to Lenz's law, the direction of $M^\prime$ is such that it opposes approaching magnet (see figure). The potential energy of the loop having magnetic moment $M^\prime$ and placed in magnetic field $\vec{B}$ is, \begin{align} U&=-\vec{M^\prime}\cdot \vec{B} \nonumber\\ &=M^\prime B\nonumber\\ &=\frac{3\pi\mu_0 a^4Mv}{2Rx^4}\,\frac{\mu_0}{4\pi}\frac{2M}{x^3}\nonumber\\ &=\frac{3}{4}\frac{\mu_0^2a^4M^2v}{Rx^7}, \end{align} and the force acting on the loop is, \begin{align} F&=-\frac{\mathrm{d}U}{\mathrm{d}x}\nonumber\\ &=\frac{21}{4}\frac{\mu_0^2a^4M^2v}{Rx^8}. \nonumber \end{align} The positive sign confirms the repulsive nature of force.